What is the area of a quadrilateral with vertices at (1, 1), (4, 1), (7, 5) and (4, 5)?

Did you notice that points (1,1) and (4,1) form a horizontal line, as do (7,5) and (4,5) ??

So if we use the first two points to form the base, the base has length 3, and the height between the two parallel lines is 4
So the area is 3x4 or 12 square units.

This was a lucky case.

To find the area of a quadrilateral, you can divide it into two triangles and calculate the area of each triangle separately.

Step 1: Calculate the length of one side of the quadrilateral.
- Start by finding the length of the horizontal side. In this case, the horizontal side is between the points (1, 1) and (4, 1).
- To find the length, subtract the x-coordinates of the two points: 4 - 1 = 3.

Step 2: Calculate the length of the vertical side.
- The vertical side is between the points (7, 5) and (4, 5). Since the y-coordinates are the same, the length of the vertical side is 0.

Step 3: Calculate the area of the first triangle:
- The first triangle is formed by the vertices (1, 1), (4, 1), and (4, 5).
- The length of the base of this triangle is the same as the length of the horizontal side we found in step 1: 3.
- To find the height of the triangle, subtract the y-coordinate of the top vertex from the y-coordinate of the bottom vertex: 5 - 1 = 4.
- Now, calculate the area of the triangle using the formula: Area = (base * height) / 2. So, (3 * 4) / 2 = 6.

Step 4: Calculate the area of the second triangle:
- The second triangle is formed by the vertices (4, 1), (7, 5), and (4, 5).
- The base length of this triangle is the same as the length of the vertical side we found in step 2: 0.
- The height of this triangle is the same as the height of the first triangle: 4.
- Using the formula: Area = (base * height) / 2, we get (0 * 4) / 2 = 0.

Step 5: Add the areas of the two triangles together to find the total area of the quadrilateral:
- 6 + 0 = 6.

Therefore, the area of the quadrilateral with vertices at (1, 1), (4, 1), (7, 5), and (4, 5) is 6 square units.